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प्रश्न
If A(4, 3), B(-1, y) and C(3, 4) are the vertices of a right triangle ABC, right-angled at A, then find the value of y.
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उत्तर
Given that A(4, 3), B(-1, y) and C(3, 4) are the vertices of the ΔABC.
ΔABC is a right triangle at A.
Hence by applying the Pythagoras Theorem, we have,
AB2 + AC2 = BC2 ....(1)
Let us find the distances, AB, BC and CA using the
distance formula.
`AB=sqrt((-1-4)^2+(y-3)^2)`
`BC=sqrt((3+1)^2+(4-y)^2)`
`CA=sqrt((3-4)^2+(4-3)^2)=sqrt2`
Squaring both the sides, we have
`AB^2=25+y^2+9-6y`
`BC^2=4+16+y^2-8y`
`AC^2=2`
Therefore, from equation (1), we have,
`25+y^2+9-6y+2=4+16+y^2-8y`
`36+y^2-6y=20+y^2-8y`
16-6y=-8y
16=-8y+6y
-y=16/2
y=-8
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Case Study -2
A hockey field is the playing surface for the game of hockey. Historically, the game was played on natural turf (grass) but nowadays it is predominantly played on an artificial turf.
It is rectangular in shape - 100 yards by 60 yards. Goals consist of two upright posts placed equidistant from the centre of the backline, joined at the top by a horizontal crossbar. The inner edges of the posts must be 3.66 metres (4 yards) apart, and the lower edge of the crossbar must be 2.14 metres (7 feet) above the ground.
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